HEX OF MATHEMATICS 4Xow it is very remarkable that Jerrard's equation lends it- self with the greatest ease to this research, and is, in the sense which we shall explain, susceptible of an actual analytic soh- tion. For we may indeed conceive the question of the algebraic solution of equations from a point of view different from that which for long has been indicated by the solution of equations of the first four degrees, and to which we are especially com- mitted. 'Instead of expressing the closely interconnected system of roots, considered as functions of the coefficients, by a formula involving many-valued radicals,* we may seek to obtain the roots expressed separately by as many distinct uniform [one- valued] functions of auxiliary variables, as in the case of the third degree. In this case, where the equation a?3 — 3# -f 2a = 0 is under discussion, it suffices, as we know, to represent the coefficient a by the sine of an angle, say A, in order that the roots be isolated as the following well-determined functions .A . a 4- 2ir . A -f 4^ 2 sin -, 2 sin —:-----, 2 sin---------. 3 3 3 [Hermite is here recalling the familiar ^trigonometric solution' of the cubic usually discussed hi the second course of school algebra. The ^auxiliary variable* is A\ the 'uniform functions* are here sines.] *Now it is an entirely similar fact which we have to exhibit concerning the equation a?5 — x — a = 0. Only, instead of sines or cosines, it is the elliptic functions which it is necessary to introduce. .,.' * For example, as in the simple quadratic a2 — a = 0: the roots are us as 4- Vcft and a? = — Va; the 'many-valuedness* of the radical involved, here a square root, or irrationality of the second degree, appears in the double sign, =fc, when we say briefly that the two roots are Va. The formula giving the three roots of cubic equations involves the three-valued irrationality j^ which has the three values 1, i( - l+V - 3), i( - 1 508